state the relations of a set

Question

Consider the following relations:

•   ⊆ on P(U) for a given universe U.

•   P is the relation follows-as-president-of-the-US between U.S. presidents. 
For instance, 〈Lyndon B. Johnson, John F. Kennedy〉∈ P and 〈George W. Bush, 
John F. Kennedy〉∈ P, since Lyndon B. Johnson and George W. Bush were presidents  
of the U.S after John F. Kennedy was.

•   T ⊆ N x N; 〈x, y〉 ∈ T if x is a factor of y, i.e. y can be divided by x 
without remainder. For instance, 〈3, 12〉 ∈ T and 〈4, 12〉 ∈T, but 〈5, 12〉 ∉ T.

Proof that ⊆, P and T are order relations and state whether they are weak or strict, total or partial.

My solution is

P -irreflexive, intransitive, asymetric, strict, partial order.
T - reflexive, transitive, antisymmetric, strict, parial order.

Can please somebody check and correct it? And also explain how to do the first statement (⊆ on P(U) for a given universe U)?

Answer 1

Hint

In general, a (partially) ordered set (or poset) $P$ is a set wuth a relation $R$ on $P$ such that $R$ is reflexive, antisymmetric, and transitive, i.e. for all $a, b, c \in P$, we have that :

$aRa$ (reflexivity)

if $aRb$ and $bRa$, then $a = b$ (antisymmetry)

if $aRb$ and $bRc$, then $aRc$ (transitivity).

To show that, for a given universe $U$, the relation $\subseteq$ is an order relation on the power-set $\mathcal P(U)$, we have to show that the above properties apply, i.e. that :

for all $A, B, C \in \mathcal P(U)$, i.e. for all $A, B, C \subseteq U$,

$A \subseteq A$ --- obvious

if $A \subseteq B$ and $B \subseteq A$, then $A = B$

if $A \subseteq B$ and $B \subseteq C$, then $A \subseteq C$.

You have to prove them from the definition of $\subseteq$ :

$A \subseteq B$ iff for all $x$ : if $x \in A$, then $x \in B$.